Page 52 - Percolation Models for Transport in Porous Media With

44 CHAPTER 3. PERCOLATION MODEL OF FLUID FLOW

After substituting q(r11 G) into (3. 7) and differentiating the obtained implicit
function with respect to r1 for a fixed G, we get

(3.9)

Let
8Gf8rt < 0, 8Gf8q > 0. (3.10)
The first inequality in (3.10) means that the greater the fraction of thin capil-
laries in a chain (the smaller r1, the greater this fraction), the greater the pressure
gradient required in the r1-chain to let a fixed flow q pass through this chain. The
second inequality in (3.10) means that the greater the flow q to pass through a
fixed r1-chain, the greater the average pressure gradient required in the chain. It
now follows from (3.9) and (3.10) that 8q(rt,G)/8rt > 0 and from (3.8), that
8k(rt, G) G-t 8q(rt, G) 0 (3.11)
8rt = J.L 8rt > ·

According to (3.11), the conductivities of the r1-chains increase with the in-
crease of r1, and therefore for every pressure gradient G, the hierarchy of r1-chains
built with respect to the radii r1 of the thinnest capillaries in the chains is the
same as the hierarchy of the chains built with respect to the average conductivities
k(rt,F).
The described approach to the construction of the laws describing the fluid flow
in porous media holds for gas flow as well, and also allows some modification for
other types of media, e.g., fractured and cavernous ones. If additional conditions
like (1.10) are satisfied, then (3.11) holds as well.
Thus the considered plan for the construction of the law describing the fluid
flow in micro heterogeneous media allows to study the flow of anomalous fluids as
well as Newtonian ones, if the laws describing the flow of the former at the micro
level are known.
To classify these laws, introduce the Reynolds number Re and the average
velocity v in the capillary of radius r, when the flow of the fluid with density Pt
equals q, as is customary

Depending on Re, the conditions of the flow in capillaries can be broken up
in three groups. When Re< Ret the flow is laminar; when Re1 $ Re $ Re2
it is transient from laminar to turbulent; when Re< Re2 it is turbulent. The
critical values Ret and Re2 of the Reynolds number for the tubes (capillaries)
with circular cross-section lie in the following intervals, 1500 < Re1 < 2100 and
1900 < Re2 < 3000 (53).